Shock waves in dilute bubbly liquids

The propagation of weak shock waves in liquids containing a small concentration of gas bubbles is studied theoretically on the basis of a mathematical model that contains all - and only - the effects that contribute to first order in the gas volume fraction. In particular, the thermal exchange between the gas bubbles and the liquid is described accurately. This aspect of the theory emerges as its most significant component, relegating effects such as the relative motion between the phases to roles of minor importance. Comparison with experimental results substantiates the accuracy of the model for shock waves that have had time to broaden from an initial sharp front to a more diffuse profile. For shock waves closer to inception, marked differences are found between theory and experiment. The same problem affects all other published theoretical treatments. It is concluded that some as yet poorly understood mechanism governs the early-time behaviour of shock waves in bubbly liquids.

[1]  Andrea Prosperetti,et al.  Nonlinear bubble dynamics , 1988 .

[2]  R. Nigmatulin.  Mathematical modelling of bubbly liquid motion and hydrodynamical effects in wave propagation phenomenon , 1982 .

[3]  L. van Wijngaarden,et al.  One-Dimensional Flow of Liquids Containing Small Gas Bubbles , 1972 .

[4]  L. Wijngaarden,et al.  Relaxation effects, caused by relative motion, on shock waves in gas-bubble/liquid mixtures , 1974, Journal of Fluid Mechanics.

[5]  Robert I. Nigmatulin,et al.  Heat exchange between a gas bubble and a liquid , 1974 .

[6]  Andrea Prosperetti,et al.  Bubble dynamics in a compressible liquid. Part 1. First-order theory , 1986, Journal of Fluid Mechanics.

[7]  L. Wijngaarden.  On the structure of shock waves in liquid-bubble mixtures , 1970 .

[8]  R. Nigmatulin,et al.  Structure of shock waves in a liquid containing gas bubbles , 1974 .

[9]  A. Prosperetti,et al.  Bubble Dynamics and Cavitation , 1977 .

[10]  Robert I. Nigmatulin,et al.  Dynamics, heat and mass transfer of vapour-gas bubbles in a liquid , 1981 .

[11]  S. Bankoff,et al.  Strong shock waves propagating through a bubbly mixture , 1984 .

[12]  R. Nigmatulin,et al.  Dynamics of vapor-gas bubbles , 1976 .

[13]  Michael J. Miksis,et al.  Effective equations for wave propagation in bubbly liquids , 1985, Journal of Fluid Mechanics.

[14]  H. G. Flynn.  Cavitation dynamics. I. A mathematical formulation , 1975 .

[15]  L. van Wijngaarden,et al.  On the equations of motion for mixtures of liquid and gas bubbles , 1968, Journal of Fluid Mechanics.

[16]  I. J. Campbell,et al.  Shock waves in a liquid containing gas bubbles , 1958, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[17]  G. Whitham,et al.  Linear and Nonlinear Waves , 1976 .

[18]  J. Rubinstein.  Bubble interaction effects on waves in bubbly liquids , 1985 .

[19]  Andrea Prosperetti,et al.  A theoretical study of sonoluminescence , 1993 .

[20]  M. Kameda,et al.  Structure of Shock Waves in a Liquid Containing Gas Bubbles , 1995 .

[21]  Andrea Prosperetti,et al.  Bubble oscillations in the nearly adiabatic limit , 1992 .

[22]  V. Kedrinskii.  Propagation of perturbations in a liquid containing gas bubbles , 1968 .

[23]  V. K. Kedrinskii,et al.  Shock waves in a liquid containing gas bubbles , 1980 .

[24]  Joseph B. Keller,et al.  Damping of Underwater Explosion Bubble Oscillations , 1956 .

[25]  Andrea Prosperetti,et al.  Linear pressure waves in bubbly liquids: Comparison between theory and experiments , 1989 .

[26]  C. Devin,et al.  SURVEY OF THERMAL, RADIATION, AND VISCOUS DAMPING OF PULSATING AIR BUBBLES IN WATER , 1959 .

[27]  Andrea Prosperetti,et al.  Bubble phenomena in sound fields: part one , 1984 .

[28]  Andrea Prosperetti,et al.  Active and passive acoustic behavior of bubble clouds at the ocean’s surface , 1993 .

[29]  Michael J. Miksis,et al.  Bubble Oscillations of Large Amplitude , 1980 .

[30]  Ali Gülhan,et al.  On the structure of nonlinear waves in liquids with gas bubbles , 1990 .

[31]  S. Bankoff,et al.  Propagation of pressure waves in bubbly mixtures , 1984 .

[32]  M. Kipp,et al.  Transient wave propagation in bubbly liquids , 1979, Journal of Fluid Mechanics.

[33]  L. Ting,et al.  Nonlinear radial oscillations of a gas bubble including thermal effects. , 1984 .

[34]  V. Kuznetsov,et al.  Propagation of perturbations in a gas-liquid mixture , 1978, Journal of Fluid Mechanics.

[35]  Andrea Prosperetti,et al.  The thermal behaviour of oscillating gas bubbles , 1991, Journal of Fluid Mechanics.

[36]  V. Shagapov,et al.  Shock structure in a liquid containing gas bubbles with nonsteady inter phase heat transfer , 1977 .

[37]  R. F. Warming,et al.  An implicit finite-difference algorithm for hyperbolic systems in conservation-law form. [application to Eulerian gasdynamic equations , 1976 .

[38]  G. Sod.  A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws , 1978 .

[39]  Andrea Prosperetti,et al.  Numerical integration methods in gas‐bubble dynamics , 1989 .